3.12 \(\int \frac{1}{\sinh ^{\frac{3}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=76 \[ -\frac{2 \cosh (a+b x)}{b \sqrt{\sinh (a+b x)}}-\frac{2 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)}} \]

[Out]

(-2*Cosh[a + b*x])/(b*Sqrt[Sinh[a + b*x]]) - ((2*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[Sinh[a + b*x]])/
(b*Sqrt[I*Sinh[a + b*x]])

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Rubi [A]  time = 0.028671, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2636, 2640, 2639} \[ -\frac{2 \cosh (a+b x)}{b \sqrt{\sinh (a+b x)}}-\frac{2 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^(-3/2),x]

[Out]

(-2*Cosh[a + b*x])/(b*Sqrt[Sinh[a + b*x]]) - ((2*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[Sinh[a + b*x]])/
(b*Sqrt[I*Sinh[a + b*x]])

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sinh ^{\frac{3}{2}}(a+b x)} \, dx &=-\frac{2 \cosh (a+b x)}{b \sqrt{\sinh (a+b x)}}+\int \sqrt{\sinh (a+b x)} \, dx\\ &=-\frac{2 \cosh (a+b x)}{b \sqrt{\sinh (a+b x)}}+\frac{\sqrt{\sinh (a+b x)} \int \sqrt{i \sinh (a+b x)} \, dx}{\sqrt{i \sinh (a+b x)}}\\ &=-\frac{2 \cosh (a+b x)}{b \sqrt{\sinh (a+b x)}}-\frac{2 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{\sinh (a+b x)}}{b \sqrt{i \sinh (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0516042, size = 57, normalized size = 0.75 \[ -\frac{2 \left (\cosh (a+b x)-\sqrt{i \sinh (a+b x)} E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{b \sqrt{\sinh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^(-3/2),x]

[Out]

(-2*(Cosh[a + b*x] - EllipticE[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*Sqrt[I*Sinh[a + b*x]]))/(b*Sqrt[Sinh[a + b*x]
])

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Maple [A]  time = 0.039, size = 154, normalized size = 2. \begin{align*}{\frac{1}{b\cosh \left ( bx+a \right ) } \left ( 2\,\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },1/2\,\sqrt{2} \right ) -\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(b*x+a)^(3/2),x)

[Out]

(2*(1-I*sinh(b*x+a))^(1/2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+a))^(1/2)*EllipticE((1-I*sinh(b*x+a))^(
1/2),1/2*2^(1/2))-(1-I*sinh(b*x+a))^(1/2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+a))^(1/2)*EllipticF((1-I
*sinh(b*x+a))^(1/2),1/2*2^(1/2))-2*cosh(b*x+a)^2)/cosh(b*x+a)/sinh(b*x+a)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(sinh(b*x + a)^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sinh \left (b x + a\right )^{\frac{3}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

integral(sinh(b*x + a)^(-3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh ^{\frac{3}{2}}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(b*x+a)**(3/2),x)

[Out]

Integral(sinh(a + b*x)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(sinh(b*x + a)^(-3/2), x)